Module-1.1-Concepts-of-Probability-S.pdf

1 ©2017 Batangas State University
INSTRUCTIONAL MATERIAL/MODULE
Engineering Data Analysis

MODULE 1
Definition of Statistical Concepts
and Principles

LEARNING OBJECTIVES
• To understand the basic concepts
used in statistics and to analyze the
basic terms in probability.

Basic Concepts of Probability
Probability is a chance of something will happen.
Definition 1: The set of all possible outcomes of a statistical
experiment is called the sample space and is represented by the
symbol S. Each outcome in a sample space is called an element, or a
member of the sample space, or simply a sample point.
If a sample space has a finite number of elements, we may list the
members separated by commas and enclosed in braces. Thus, the
sample space S, of possible outcomes when a die is tossed may be
written }
6
,
5
,
4
,
3
,
2
,
1
{

S

Example 1: Consider an experiment of flipping a coin, what are
the possible outcomes?
S = {H,T}

Example 2: An experiment consists of flipping a coin and then
flipping it a second time if a head occurs. If a tail occurs on the
first flip, then a die is tossed once. List the possible sample
points.

Example 3: Suppose that three items are selected at random from a
manufacturing process. Each item is inspected and classified defective D, or
non-defective N. List the elements of the sample space.

Sample space with a large or infinite number of sample points
are best described by a statement or rule method. For example,
if the possible outcomes of an experiment are the set of cities in
the world with a population over 1 million, our sample space is
written
is a city with a population over 1 million}
If S is a set of all points (x,y) on the boundary or the interior of a
circle of radius 2 with center at the origin, we write the rule
x
x
S {

}
4
2
2
)
,
{( 

 y
x
y
x
S

Definition 2: An event is a subset of a sample space.
Given the sample space ,where t is the life in years of a
certain electronic component, then the event A that the
component fails before the end of the fifth year is the subset
Definition 3: The complement of an event A with respect to S is
a subset of all element of S that are not in A. We denote the
complement of A by the symbol A’.
}
0
{ 
 t
t
S
}
5
0
{ 

 t
t
A

Consider the sample space
Let A = {book, stationery, laptop, paper}
Then the complement of A is A’ = {cellphone, mp3}
Definition 4: The intersection of two events A and B denoted by the symbol
A∩B, is the event containing all elements that are common to A and B.
Let E be the event that a person selected at random in a classroom is
majoring in engineering, and let F be the event that the person is female.
Then E∩F is the event of all female engineering students in the classroom.
}
,
,
,
3
,
,
{ laptop
stationery
paper
mp
cellphone
book
S 

Let V = {a,e,i,o,u) and C = {l,r,s,t} then it follows that V∩C =Ф,
that is, if A and B have no elements in common.
For certain statistical experiment, it is by no means unusual to
define two events A and B, and cannot both occur simultaneously.
The events A and B are the said to be mutually exclusive. Stated
more formally, we have the following definition.
Definition 5: Two events A and B are mutually exclusive, or
disjoint, if A∩B = Ф, that is, if A and B have no elements in
common.

Definition 6: The union of the two events A and B, denoted by the
symbol AUB, is the event containing all the elements that belong to A
or B or both.
Example: Let A = {a,b,c} and B = {b,c,d,e}; then A U B = {a,b,c,d,e}
Example: Let P be the event that an employee selected at random from
an oil drilling company smokes cigarettes. Let Q be the event that the
employee selected drinks alcoholic beverages. Then the event PUQ is
the set of all employees who either drink or smoke or do both.

A
7
B
2
6
1
4 3
5
C
1. Determine the following:
a) A U C
b) B’∩A
c) A∩B∩C
d) (AUB)∩C’

Assessment Task: Basic Concepts of Probability
1. List the elements of each of the following sample spaces:
a) the set of integers between 1 and 50 divisible by 8
b) the set + 4x -5 =0}
c) the set of outcomes when a coin is tossed until a tail or three heads
appear
d) the set is a continent}
2. An experiment consists of tossing a die and then flipping a coin
once if the number on the die is even. If the number on the die is odd,
the coin is flipped twice. List down the sample points.
2
{ x
x
S 
x
x
S {


Counting Sample Points
Theorem 1 : If an operation can be performed in
n1 ways, and if for each of these ways a second
operation can be performed in n2 ways, then the
two operations can be performed together in n1n2
ways.
Example: How many sample points are there in
the sample space when a pair of dice is thrown
once?
N1= 6
N2=6
N1N2 = 6(6) = 36 samples

Example: A developer of a new subdivision
offers prospective home buyers a choice of Tudor,
Rustic, Colonial and Traditional exterior styling
in ranch, two-story, and split level floor plans. In
how many different ways can a buyer order one
of these homes?
Example: If a 22-member club needs to elect a
chair and a treasurer, how many different ways
can these two to be elected?

Theorem 2: If an operation can be performed in n1
ways, and if for each of these a second operation can
be performed in n2 ways, and for each of the first two
a third operation can be performed in n3 ways, and so
forth, then the sequence of k operations can be
performed in n1n2...nk ways.
Example: Sam is going to assemble a computer by
himself. He has the choice of chips from two brands,
a hard drive from four, memory from three, and an
accessory bundle from five local stores. How many
different ways can Sam order the parts?
N1=2 n1n2n3n4= 2(4)(3)(5) = 120 ways
N2=4
N3=3
N4=5

Example: A drug for the relief of asthma can be
purchased from 5 different manufacturers in
liquid, tablet, or capsule form, all of which come
in regular and extra strength. How many different
ways can a doctor prescribe the drug for a patient
suffering from asthma?

Theorem 3: A permutation is an arrangement
of all or part of a set of objects. The number of
permutation of n objects is n!
Example: In how many ways can 5 examinees
be lined up to go inside the testing centers?
5! = 120 ways
Example: In how many ways can four dating
reviewees be seated in the review center
without restriction?

Theorem 4: The number of permutations of n
distinct objects taken r at a time is
, r ≤ n
Example: In one year, three awards (research,
teaching ans service) will be given to a class of 25
graduate students in a statistics department. If
each student can receive at most one award, how
many possible selections are there?
)!
(
!
Pr
r
n
n
n


nPr = permutations
n= total number of objects
r = number of objects selected

Theorem 5: The number of permutations of n
object arranged in a circle is (n-1)! Permutation
that occur by arranging objects in a circle are called
circular permutations.
Example: In how many ways can 6 students be
seated in a round dining table?
(6-1)! = 120 ways
Example: Ten boy scouts are seated around a camp
fire. How many ways can they be arranged?

Theorem 6: The number of distinct permutations of n
things of which n1 are of one kind, n2 of a second
kind,..., nk of a kth kind is
Example: In a college football training session, the
defensive coordinator needs to have 10 players
standing in a row. Among these 10 players, there are 1
freshman, 2 sophomores, 4 juniors and 3 seniors. How
many different ways can they be arranged in a row if
only their class level will be distinguished?
!
!...
2
!
1
!
nk
n
n
n

Theorem 7: The number of ways of partitioning a
set of n objects into r cells with n1 elements in the
first cell, n2 elements in the second, and so forth, is
Example: In how many ways can 7 graduate
students be assigned to 1 triple and 2 double hotel
rooms during a conference?
!
!...
2
!
1
!
nr
n
n
n

Theorem 8: The number of combinations of n
distinct objects taken r at a time is
Example: How many ways are there to select 3
applicants from 8 equally qualified Engineers
for a Staff Engineer position in a
Semiconductor company.
)!
(
!
!
r
n
r
n


Probability of an Event
The probability of an event A is the sum of the weights of all sample
points in A. Therefore,
P(Ф) = 0 P(S) = 1
Furthermore, if A1,A2,A3,... is a sequence of mutually exclusive
events, then
P(A1UA2UA3) = P(A1) + P(A2) + P(A3) + ...
1
)
(
0 
 A
P

26
©2017 Batangas State University
Example: A coin is tossed twice. What is the
probability that at least 1 head occurs?
Example: A die is loaded in such a way that an
even number is twice as likely to occur as an odd
number. If E is the event that a number less than 4
occurs on a single toss of the die, find P(E).

Theorem 9: If an experiment can result in any of
N different equally likely outcomes, and if exactly
n of these outcomes correspond to event A, then
the probability of event A is
N
n
A
P 
)
(

Example: A statistics class for engineers consists of
25 industrial, 10 petroleum, 10 electrical and 8
sanitary engineering students. If a person is
randomly selected by the instructor to answer a
question, find the probability that the student chosen
is (a) an industrial engineering major (b) petroleum
engineering or an sanitary engineering major.

Module-1.1-Concepts-of-Probability-S.pdf

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